1

1 TABLE OF CONTENTS

2 Introduction ........................................................................................................................ 3

2.1 Introduction and Literature Review ............................................................................ 3

2.2 Project Aims ................................................................................................................ 5

3 System Characterisation..................................................................................................... 6

3.1 Instrumentation and Apparatus ................................................................................... 6

3.1.1 Inherited ............................................................................................................... 6

3.1.2 Updates ................................................................................................................ 7

3.2 Disc Assembly Characterisation ................................................................................. 8

3.2.1 Disc ...................................................................................................................... 9

3.2.2 Whole Assembly ................................................................................................ 12

3.3 Pin Assembly Characterisation ................................................................................. 14

3.3.1 Analytic .............................................................................................................. 14

3.3.2 Experimental ...................................................................................................... 14

4 System Calibration ........................................................................................................... 18

4.1 Dynamometer ............................................................................................................ 18

4.2 Accelerometers .......................................................................................................... 18

4.3 Force Transducer ....................................................................................................... 20

4.3.1 Method by Weights ............................................................................................ 20

4.3.2 Method by Wire ................................................................................................. 21

4.4 Mass Compensation .................................................................................................. 22

4.4.1 First Estimate ..................................................................................................... 22

4.4.2 Method of Least Squares ................................................................................... 23

5 Bulk Data Analysis .......................................................................................................... 25

5.1 Methods ..................................................................................................................... 25

5.1.1 Measurement ...................................................................................................... 25

5.1.2 Post-Processing .................................................................................................. 27

5.1.3 Squeal Frequency Analysis ................................................................................ 28

5.1.4 Fixed In Space Theory ....................................................................................... 32

6 Case Studies ..................................................................................................................... 34

6.1 The Importance of Radial Fluctuations ..................................................................... 34

6.2 Comparisons with Coulomb Friction ........................................................................ 36

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6.2.1 Method ............................................................................................................... 36

6.2.2 Backwards Rotating Squeal ............................................................................... 38

6.2.3 Forwards Rotating Squeal .................................................................................. 42

6.3 Limit Cycle Behaviour .............................................................................................. 44

7 Conclusions ...................................................................................................................... 47

8 Appendix .......................................................................................................................... 48

8.1 Risk Assessment ........................................................................................................ 48

9 References ........................................................................................................................ 49

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2 INTRODUCTION

2.1 INTRODUCTION AND LITERATURE REVIEW

Squeal can occur in any disc brake system that exhibits friction-induced vibration. It is the

noise produced when such a system resonates, and it causes problems in the automotive

industry mostly due to the harshness of the noise to the human ear, but also due to the

unpredictable wear and occasionally damage caused to brake systems. Returns on vehicles

under warranty due to the noise nuisance are common, and are predicted to generate warranty

costs in the region of $1 billion (US) a year in North America alone [1]. It is therefore still a

universally poorly understood phenomenon, and research efforts are ongoing to understand,

predict and alleviate the problem. This project aimed to undertake an experimental

investigation on a reduced pin-on-disc model of a disc brake.

Due to the complex and twitchy nature of squeal, the literature is vast, and can be divided very

approximately into structural modelling efforts, friction modelling, linear, non-linear and

experimental studies. Structural modelling and experimentation studies explore the effects of

modifications made to the structural elements of a disc brake setup. For example, Wagner et

al. [2] investigated the structural optimisation of an automotive brake disc using asymmetric

cooling channels to split the eigenfrequencies of the system to avoid squeal, with successful

numerical and experimental results for a specific disc and pad geometry. Hammersmith and

Jacobson [3] employed a spiral shaped modification of the disc brake surface topography with

a similar aim; here the results showed that the modifications are only effective in the short term.

There are various models of friction which are studied in the literature, and are summarised

well by Woodhouse et al. [4]. The “friction-curve” and “rate-and-state” model types are

discussed, as is the necessity for reliable constitutive models of friction in effective numerical

predictions based on a dynamic friction contact, particularly relevant in the problem of disc

brake squeal.

Linear studies are vast in the literature, for example the work of Hoffman et al. [5], where a 2

degree of freedom system is employed to investigate the physical mechanisms underlying the

mode coupling instability. Similarities are drawn to the well-known problem of flutter in

aeroelasticity, and conditions for cyclic growth of vibrational energy proposed. Butlin and

Woodhouse [6] performed sensitivity studies using a linear approach, and regarded the

twitchiness of squeal as an important part of the phenomenon, rather than an experimental

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difficulty. An experimental study on squeal initiation was also performed on a similar pin-on-

disc model which is used in this project. Linear studies with particular applications are also in

abundance. For example the work of Lorang et al. [7] on the specific application of disc brake

squeal on TGV trains in France, where a flutter instability analysis is adopted under the action

of Coulomb friction.

Non-linear studies focus on obtaining predictions of fully developed vibration states during

brake squeal, where linear methods cannot be applied. Such studies include the work of

Coudeyras et al. [8], where a harmonic balance method is adopted and limit-cycle computations

made.

Experimental investigations of squeal include the work of Giannini et al. [9], on a partially

reduced setup with a small (1 cm2) area brake made in contact on both sides of the disc (in a

conventional manner). The paper has a number of conclusions, one of which is that vibration

modes of the disc during squeal events are all fixed in space, which is an observation verified

for only one squeal frequency in this project. The nature of the clamping brake leads to

complications with detailed force measurement however, and the conclusions made about stick

slip do not account for fluctuations in disc velocity at the contact points. Another investigation

by Massi and Giannini [10] investigates the effect of damping on the propensity of squeal for

a simplified beam-on-disc model. The key finding was that the larger the ratio between the

modal damping factors of two coalescing modes, the larger the propensity of the system to

squeal. Other experimental studies attempt active squeal control, such as the work of Cunefare

and Graf [11], where control was attempted using ultrasonic dither. These studies are

interesting but again may only show to be effective for the particular system under

investigation, and understanding of squeal is overlooked in the path to prevention.

The information missing from experimental studies in the literature involves a detailed analysis

of the contact forces and dynamics from squeal initiation to limit cycle behaviour. Numerical

investigations often assume a form of friction model at the start, and all subsequent results are

the consequence of the model chosen, without gauging whether the model used is a reliable

one. An experimental investigation is required where contact forces, velocities and

displacements are able to be measured accurately, so that friction models can be validated and

the reliability of numerical predictions substantiated.

An experimental test rig was inherited for this project, which was used in investigations by

Duffour [12], Butlin [13] and Dawes [14] previously. The project ran alongside a numerical

5

investigation of brake squeal (Edmondson-Jones [15]) using the same model as a basis for

finite element modelling.

2.2 PROJECT AIMS

The project aims consisted of the following:

To add instrumentation to the current test rig in order to measure contact forces

accurately in 3 dimensions;

To characterise the dynamics of the separate pin and disc assemblies;

To perform a large number of squeal tests under various conditions, with the aim of

finding parameter correlations and dominant squeal frequencies;

To measure and analyse the importance of off-axis fluctuations during squeal;

To undertake an investigation into the contact forces, velocities and displacements

during limit cycle behaviour;

To compare measured friction forces with expectations of Coulomb friction.

The work of this project will provide a point of reference for future work on disc brake squeal,

and it is hoped that the advancement in the methods employed for detailed contact force

measurement will be continued. This will then provide a sound basis for the adoption of certain

friction models in numerical investigations of brake squeal, with the aim of reliably predicting

and alleviating the problem.

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3 SYSTEM CHARACTERISATION

In the field of linear vibration, the resulting behaviour of two systems brought into contact at a

single point can be predicted reasonably straightforwardly. With knowledge of the applied

constraint, the range of each natural frequency is also known, irrespective of the exact position

of the point of coupling (this is the theory of “interlacing”, revealed by Rayleigh in 1894 [16]).

During steady sliding at the contact one might assume that this theory may still be applicable.

However, in the presence of friction, instabilities can grow to give non-linear squeal behaviour.

Knowledge of the system dynamics forms the basis of any squeal investigation, and plays

particular importance in the interpretation of measured parameters, seen later in the Case

Studies section.

3.1 INSTRUMENTATION AND APPARATUS

3.1.1 Inherited

The apparatus used was based on previous departmental projects investigating disc brake

squeal, and can be split into two systems. The disc assembly features an aluminium disc of

radius 130 mm and thickness 11 mm which is attached through a shaft to a rotor. This

provides the system with significantly more rotational inertia and something to grasp in order

to spin the disc. The assembly is mounted in two bearing sets as shown in Figure 1 below:

Figure 1. Labelled disc assembly.

The pin assembly consists of a heavy support section which is mounted on leaf springs and

controlled by a preloading coil spring. This support section has a sub frame onto which an

aluminium dynamometer shaped like a top hat is attached. The dynamometer has a thin walled

cylindrical section onto which strain gauges are glued, with a flat top face where the “pin” is

fixed. The pin is actually a hemispherical polycarbonate tip, and is called a pin with respect to

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the fact that we may look to simplify contact pressure distributions to forces at a single point.

Polycarbonate is isotropic, and has a reasonably high friction coefficient in contact with the

aluminium disc, to create a system that will readily squeal. Figure 2 shows a detailed diagram

of the top hat dynamometer taken from Butlin & Woodhouse [17].

Figure 2. Diagram of part of the pin assembly, taken from [17].

3.1.2 Updates

One of the original goals of the project was to re-instrument the pin system to enable more

accurate and reliable measurement of forces at the contact between pin and disc, in all three

dimensions. The dynamometer provides a good method for measuring DC and low frequency

variations in the friction and normal forces, but for higher frequency applications the signal is

noisy. A low pass filtered dynamometer signal coupled with a force transducer (which uses

piezoelectric crystals as a means of measurement), provides a good method for accurately

measuring contact forces. The force transducer chosen was a PCB 260A11, and is shown in its

mounted position in Figure 3. The axes used to clarify directions of measurement in this report

are based on the orthogonal set printed on the force transducer, and added in Figure 3. Hence,

a positive normal force corresponds to a positive 𝑧 force, and the friction force is measured as

positive in the positive 𝑦 direction. Other terminology used frequently in this report relates to

the direction of rotation of the disc; what will be described as “forward” rotation is clockwise

rotation when observing the disc assembly from the right hand side of Figure 1, and gives rise

to positive net measured friction forces. “Backwards” rotation corresponds to anti-clockwise

rotation, and to negative net measured friction forces.

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Figure 3. Labelled pin assembly after force transducer addition.

Other measurement apparatus used included 2 smaller accelerometers, allowing 3 axes of

acceleration measurement for the pin tip in conjunction with a normal accelerometer inside the

dynamometer (clarified in Figure 3), as well as slip rings servicing an accelerometer fixed to

the rear of the disc surface for disc vibration measurement. To measure the position and speed

of the disc, a ring of 120 equally spaced holes is mounted on the disc assembly shaft, which is

used in conjunction with an optical switch to provide positional accuracy to ±1.5°. Amplifiers

and filters were used in conjunction with the sensors, and an 8 channel data acquisition card

used to capture these signals which were recorded in Matlab using a data logging program

developed by J. Woodhouse [18].

3.2 DISC ASSEMBLY CHARACTERISATION

In order to undertake an investigation into squeal, it is necessary to fully understand the

mechanical characteristics of the apparatus being used. This is so that dominant frequencies

measured during squeal events can be compared to resonant frequencies of the separate

systems, to draw conclusions on the influencing factors of certain squeal regimes. Comparisons

with results of numerical predictions of the system behaviour are also valuable, made possible

by the work of Edmondson-Jones [15] for a parallel 4th year project. The disc assembly

dynamics will be altered by the disc, rotor and shaft rotation, due to centrifugal stiffening, an

effect studied by Lamb and Southwell [19]. The rotation speeds in the experiments undertaken

were very small, so this effect was assumed to be negligible. The previous pin assembly

dynamics were altered due to the addition of the force transducer, making comparisons of the

pin dynamics before and after this addition possible.

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3.2.1 Disc

Modes of interest of the disc assembly are either pure disc modes, or overall assembly modes.

Pure disc modes are characterised by either transverse or in-plane motion. Experimental

measurement and analytic modelling were used to gain insight into these transverse modes,

and comparisons were made with results from finite element modelling performed by

Edmondson-Jones [15]. Elementary insight into the overall assembly dynamics was gained

through modal tests.

3.2.1.1 Analytic

The analytic modelling of the disc deals only with transverse disc modes, and assumes Euler-

Bernoulli theory is valid, which is to say that the shear strain experienced in vibration is

negligible compared to the out of plane bending. The disc is modelled as having uniform

thickness of 11 mm , outer radius 130 mm and an inner clamped radius of 38 mm . This

represents a simplification of the actual inner boundary condition; the disc has three bolts which

fasten it to an aluminium block, where the block has radius 38 mm . The details of the

calculation of the disc modes have been omitted here, as they follow closely the steps outlined

by Duffour [12] on a disc with similar dimensions but with a smaller inner clamped radius. The

problem is a common one in linear vibration theory, and makes use of a separation of variables

(of radius, angle and time), where the angular dependence is a sinusoidal one and the radial has

solutions in the form of Bessel functions. The boundary conditions at the fixture radius are that

the transverse displacement and change in transverse displacement are both zero, while at the

outer radius the shear force and bending moment are zero. The governing equation of motion

for plate vibration in polar coordinates is:

𝜕2𝑤

𝜕𝑟2+

1

𝑟

𝜕𝑤

𝜕𝑟+

1

𝑟2

𝜕2𝑤

𝜕𝜃2+

𝑚

𝐷

𝜕2𝑤

𝜕𝑡2= 0

𝑤 = 𝑓(𝑟, 𝜃, 𝑡) ; 𝐷 =𝐸ℎ3

12(1 − 𝜈2)

Where 𝑤 is the transverse displacement, 𝑟 the radius, 𝜃 the angle with respect to a fixed

orientation, 𝑡 is time, 𝑚 is the mass per unit area, ℎ is the disc thickness, 𝐸 is the elastic

modulus and 𝜈 is Poisson’s ratio.

Upon assuming harmonic time dependence and separating the variables, the solution has the

form:

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𝑤(𝑟, 𝜃) = [𝐶1𝑘𝐽𝑘(𝑟) + 𝐶2𝑘𝑌𝑘(𝛽𝑟) + 𝐶3𝑘𝐼𝑘(𝛽𝑟) + 𝐶4𝑘𝐾𝑘(𝛽𝑟)] cos(𝑘𝜃 − ∅𝑘)

𝛽 = √𝜔2𝑚

𝐷

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Where 𝜔 is the angular frequency, the 𝐶𝑖𝑘 are constants determined from the radial boundary

conditions, ∅𝑘 an arbitrary angular constant, 𝐽𝑘 and 𝑌𝑘 the Bessel functions of order 𝑘 , and

𝐼𝑘and 𝐾𝑘 the modified Bessel functions of order 𝑘. Finding the coefficients involves satisfying

the 4 equations resulting from each boundary condition.

The family of resulting solutions (or modes) can be described by specifying the number of

nodal diameters and the number of nodal circles (𝑘 and 𝑛 respectively) which arise as zeros to

the angular and radial governing equations respectively. Therefore, values of 𝛽 at which zeros

to the radial part of the equation occur are denoted in ascending order as 𝛽𝑘0, 𝛽𝑘1, 𝛽𝑘2 … 𝛽𝑘𝑛

and are used to find natural frequencies of the disc.

3.2.1.2 Experimental

Transverse modes were measured by using a fixed accelerometer and tapping a grid of points

plotted on the disc. This was intended to be of sufficient resolution to measure modes up to

25 kHz, with the positional resolution of points predicted from the analytic model showing

modes with up to 2 nodal circles and 11 nodal diameters in this range. Tests were performed

at 45 different angles (8° intervals) for the required resolution but only at the contact radius for

simplicity of post-processing. More detailed modal analyses may be performed in the future

should squeal investigations necessitate it, but enough insight is gained through this resolution

here.

The method of finding specific mode shapes involved plotting all 45 transfer functions in one

plot, and analysing each resonant peak in turn. Fitting each transfer function in a window

around each peak using the Rational Fraction Polynomial method (see Richardson [20] and

[18]) allows extraction of linear vibration parameters around a resonant peak, of which the

modal amplitude factor is of interest here. Plotting the modal amplitude factors at the positions

they were gathered allows a visual interpretation of the measured mode shape. Figure 4 shows

2 examples, corresponding to modes with 2 and 3 nodal diameters, and zero nodal circles. For

visual interpretation the relative displacements have been linearly interpolated from the inner

radius (zero displacement), through the contact point radius where the data was measured, to

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the outer edge. Green within the plot indicates nodal lines, whereas antinodes pass through

positions of intense blue and yellow.

Figure 4. Left) Experimentally measured (k,n)=(2,0) disc mode. Right) Experimentally measured

(k,n)=(3,0) disc mode.

3.2.1.3 Finite Element Modelling

As a third method of verification, and in order to investigate other resonances of the disc, a

finite element model was developed. This work was carried out by Edmondson-Jones [15].

3.2.1.4 Results

The results for all three methods are summarised in Table 1, and compared for the cases of zero

nodal circles:

Experimental FEM Analytic n=0; (a) n=0; (b) (b)-(a) n=0; (c) (c)-(a) n=1 n=2

k=0 722 1040 318 213 -509 3195 12773 k=1 623 1014 391 113 -510 4353 14239 k=2 1001 1220 219 877 -124 6209 16522 k=3 2076 2057 -19 1996 -80 8737 19689 k=4 3561 3414 -147 3494 -67 11866 k=5 5392 5142 -250 5358 -34 15512 k=6 7457 7176 -281 7579 122 19617 k=7 9786 9470 -316 10150 364 k=8 12338 12004 -334 13069 731 k=9 15098 14765 -333 16333 1235

k=10 18044 17712 -332 19940 1896

Table 1. Disc resonant frequencies by Experimental, FEM and Analytical methods.

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The FEM results show reasonable agreement with the experimental results, inspiring

confidence that the modes are matched to the resonant frequencies correctly. Differences may

be due to the assumptions made about the inner boundary condition, or subtleties such as slight

variations in thickness of the disc (there is a ≈ 0.5 mm change in thickness at a radius of

69 mm) and in the material density.

Agreement with the analytic model is not so well founded. This is suspected to be a mistake in

the calculation of the Bessel function coefficients in the governing equation. In fact, 𝜕𝑤

𝜕𝑟 is not

zero at the inner radius and it should be. This was noted, but re-calculations were not

undertaken in order to maintain an experimental focus to the project.

3.2.2 Whole Assembly

In order to gain a basic knowledge of the overall assembly response, transfer functions were

measured between the rotor and the disc. These tests were designed to be simple and to give a

more complete picture of the whole system response for reference during squeal tests. Of note

here is the low frequency response from striking in the transverse direction at the top of the

rotor edge, and measuring at the disc face in a position directly below the assembly rotation

axis at the pin-disc contact radius. These positions are indicated in Figure 1. This transfer

function, displayed in Figure 5, shows low frequency resonances at 70 Hz and 130 − 140 Hz

(which are suspected to be different polarisations of a coupled rotor-disc mode). The red plot

shows the transfer function which is averaged over 10 strikes, and the blue plot gives a measure

of the coherence, or agreement between each of the measurements. This 140 Hz resonance was

also picked up in the transverse disc mode tests, where it was seen to show a form similar to

the disc mode of one nodal diameter and zero nodal circles. Figure 7 shows this peak at 140 Hz

in a plot showing the 45 transfer functions from around the disc at the contact point (as well as

the first 2 disc resonances), and Figure 6 shows the result of reconstructing this disc assembly

mode from the data.

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Figure 5. Low frequency response of the disc assembly transfer function, striking at the top rotor edge

and measuring at the disc face.

Figure 7. Transfer functions of all 45 measured

angles, showing resonance at 139Hz and the first 2

disc modes.

Figure 6. 139Hz mode shape measured at the

disc.

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3.3 PIN ASSEMBLY CHARACTERISATION

3.3.1 Analytic

The pin assembly was analysed in more detail in its inherited form (prior to the addition of the

3-axis force transducer). A lumped-parameter model developed by Dawes [14] was used to

predict the behaviour of the pin structure. This model is based on the assumption that the

dynamic behaviour of the system is dependent on the compliance of the steel bar to which the

top hat assembly is mounted. It assumes that the bar is constrained to a fixed body (the heavy

support section mentioned earlier), and that the intricate dynamic behaviour of the top hat can

be overlooked and instead idealised as a point mass with moments of inertia calculated through

finite element modelling. For more detail see Dawes [14].

The model is highly sensitive to dimensional subtleties such as the length taken for the thin

members of the mounting. These members have a rounded chamfer at each end to alleviate

stress concentrations, which poses a difficulty for choosing an effective length to use in the

model. Taking the minimum and maximum lengths of each and performing an eigenvalue

analysis yields frequencies for the two modes of vibration in the ranges 870 − 900 Hz and

2590 − 2850 Hz.

3.3.2 Experimental

Measurement of the pin system transfer function in the 𝑧 direction (say 𝑃𝑧𝑧) shows resonant

peaks at 870 Hz and 2450 Hz, which are on (and just below) the lower bound of the analytic

calculation. This may be expected due to the addition of the accelerometers which were not

accounted for in the model. Figure 8 shows 𝑃𝑥𝑥, 𝑃𝑦𝑦 and 𝑃𝑧𝑧 transfer functions of the pin up

to 4 kHz, where 𝑥, 𝑦 and 𝑧 are the directions specified in Figure 3.

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Figure 8. Initial pin system transfer functions; Pxx, Pyy and Pzz.

𝑃𝑦𝑦 and 𝑃𝑧𝑧 are very similar and both match well to the analytic calculation, since they

involve vibration in the 𝑦 – 𝑧 plane predominantly. 𝑃𝑥𝑥 describes the pure out of plane

response (with respect to the analytic model), with a clear peak at 485 Hz.

Looking at the responses at higher frequencies, the 𝑃𝑥𝑥 and 𝑃𝑦𝑦 transfer functions show a

similar peak in response at around 10 kHz, shown in Figure 9 below:

Figure 9. Initial pin system transfer functions, up to 15 kHz.

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This doesn’t show in 𝑃𝑧𝑧 which suggests it might involve bending motion of the thin walled

dynamometer. The 𝑃𝑥𝑥 transfer function may not be particularly illuminating on its own, as

we might expect that most forces experienced during braking and squeal are in the friction and

normal directions (and this is indeed verified with radial fluctuation tests in the Case Studies

section). Hence, plots of 𝑃𝑧𝑥 and 𝑃𝑦𝑥 give more insight into the out of plane structural

response of the pin system. A plot of 𝑃𝑧𝑥 is shown in Figure 10 below alongside the 𝑃𝑧𝑧 plot,

indicating that the out of plane 𝑃𝑧𝑥 peak response is 30 dB less than the 𝑃𝑥𝑥 response (from

Figure 9) measured with the same accelerometer, and therefore radial ( 𝑥 direction)

accelerations induced (by normal forces at least) will be very small.

Figure 10. Initial Pzz and Pzx transfer functions.

After the addition of the force transducer to the top hat, the natural frequencies of the system

are expected to reduce, in approximate accordance with the intersections of the force transducer

rigid body response and the previous system response, and due to the increased moment of

inertia about all axes. This is indeed the case, and the response shows a shift in the 𝑦 – 𝑧 plane

natural frequencies to 555 Hz and 1705 Hz, and a corresponding shift in the out of plane

resonance to 260 Hz, as shown in Figure 11.

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Figure 11. New pin assembly transfer functions. Old transfer functions are displayed faintly for

comparison.

The old transfer functions are displayed faintly for comparison. The new 𝑃𝑧𝑧 transfer function

no longer has an anti-resonance between the 2 resonant peaks, which is what one would expect

for a driving point response. This is a consequence of the added mass and reduction in stiffness

between the tip and 𝑧 direction accelerometer, meaning that the assumption of the tip being the

driving point is now less valid. The new 𝑃𝑧𝑥 transfer function now has a lowered response at

the new in-plane 555 Hz resonance.

All of the transfer functions displayed in this section were calculated prior to accelerometer

calibration, and so the relative magnitudes of the displayed plots measured from different

accelerometers is arbitrary.

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4 SYSTEM CALIBRATION

One of the key aims of this project was to add instrumentation to the test rig to enable better

measurement of the contact forces during squeal events.

Each sensor used required careful calibration, which is described in this section. Accurate

calibration, or at least knowledge of the level of accuracy of calibration is key in the

interpretation of experimental results. Calibration of accelerometers, strain gauges and the

force transducer were undertaken, as well as mass compensation of the force transducer.

4.1 DYNAMOMETER

Strain gauges in general have a fairly poor signal to noise ratio. They do however have the

ability to make accurate DC and low frequency measurements, which is what this project aimed

to use them for.

The previously implemented setup (used in [12], [13] and [14]) follows methods outlined by

Smith [21], where a top hat dynamometer is defined which can house multiple cross pairs of

gauges. On the current setup only 2 crossed pairs have been used for simplicity to measure the

friction and normal force.

Calibration on the strain gauges involved hanging weights from the dynamometer tip, in

orientations such that the device was loaded in the friction (𝑦) and negative normal (-𝑧)

directions separately (directions defined in Figure 3). Increments of 5 lb masses were used,

and the acceleration due to gravity taken as 9.81 ms−2. A line of best fit was then plotted for

loading and unloading in order to find the calibration factors. Equation 1 shows how the true

forces relate to the measured voltages, at a fixed gain of 1 on both gauge amplifiers:

[𝐹𝑁

𝐹𝐹𝑟] = [

−93.82 3.69−1.49 −24.65

] [𝑉𝑁

𝑉𝐹𝑟]

Equation 1.

Strain gauges are susceptible to drift and need frequent zeroing. This precaution was taken

between each test performed.

4.2 ACCELEROMETERS

Calibration of all accelerometers is required in order to calculate accurately the acceleration,

velocity and displacement at the contact, and also to apply an effective mass compensation to

19

the force transducer. Ideally, a frequency dependent calibration factor would be applied to each

accelerometer, to fully account for its frequency response in the measurable range. This may

be done using a laser vibrometer, but makes the process of calibrating a measured signal more

complex. Also, accelerometers will pick up to some extent acceleration in unintended

directions. This can be compensated for with an array of 3 orthogonal accelerometers, but

complicates any measurement and calibration of it further.

It was decided that a single calibration factor (with units ms−2 pC⁄ ) would be assigned to each

accelerometer, taken as an average of several “best fit” frequency domain windows around

resonances of the systems used for calibration, within suitable working ranges of the

accelerometers.

This process was performed initially with a laser vibrometer on a single accelerometer, which

was subsequently used to calibrate the other accelerometers. The laser vibrometer measures the

velocity of a surface, so multiplication by a factor of 𝑖𝜔 is required to give the acceleration

(assuming harmonic time dependence). This initial calibration used a lightly damped string

supported beam provided with an impulse, where the accelerometer mounting position was on

the opposite side to the incident laser beam, clarified by Figure 12:

Figure 12. Diagram showing the setup of the first accelerometer calibration using a laser vibrometer.

Subsequent accelerometers were calibrated using the disc, stacked with the calibrated

accelerometer or on the other disc face at the same planar position. The calibrated

accelerometers are described in Table 2.

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Accelerometer Brand Model Calibration factor (𝑚𝑠−2 𝑝𝐶⁄ )

Polarity; When struck on mounting face

1 Endevco 2222C 8.204 Positive

2 Endevco 2222C 7.913 Positive

3 DJB A/23/TS 1.557 Positive

4 DJB A/23/E 1.804 Positive

Table 2. Table of accelerometer information

4.3 FORCE TRANSDUCER

The force transducer was introduced as a way of improving the accuracy of force

measurements, as well as providing a method of measuring radial forces at the disc-pin contact.

The manufacturer recommends that the sensor be pre-stressed in its 𝑧 direction, which was

found not to be necessary for this application as it is achieved when the pin is preloaded onto

the disc. Calibration of the sensor involves finding the 3x3 matrix which applied to the

measured voltages gives the applied forces:

[

𝐹𝑋 𝑎𝑝𝑝𝑙𝑖𝑒𝑑

𝐹𝑌 𝑎𝑝𝑝𝑙𝑖𝑒𝑑

𝐹𝑍 𝑎𝑝𝑝𝑙𝑖𝑒𝑑

] = [

𝐶𝑥𝑥 𝐶𝑥𝑦 𝐶𝑥𝑧𝐶𝑦𝑥 𝐶𝑦𝑦 𝐶𝑦𝑧𝐶𝑧𝑥 𝐶𝑧𝑦 𝐶𝑧𝑧

] [𝑉𝑋 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

𝑉𝑌 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

𝑉𝑍 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

]

Equation 2.

4.3.1 Method by Weights

Initially a method of calibration was adopted whereby masses where added in turn to the

transducer and suddenly removed, giving a sharp change in force and hence a corresponding

voltage to calibrate. 5 tests were performed at each weight increment, and the average of each

step change in voltage taken to calibrate. Example signals are shown for a mass of 950 g in

Figure 13.

Figure 13. Two 950g calibration tests giving different off-axis results.

21

The tests were then repeated with the transducer mounted at a small known angle, in an attempt

to calibrate the other directions. This was restrictive due to the fact that the masses would start

to slide and tip at angles ≥ 12°. While allowing good calibration for the z-direction forces, this

method was found to be non-satisfactory as a means for calibrating the 𝑥 and 𝑦 directions and

the cross terms of the calibration matrix due to excessive variation in measurements, caused

predominantly by a lack of consistency from the sudden removal of weights. This is illustrated

in Figure 13 above, where the 𝑥 direction reads differently for a test under exactly the same

conditions. It is thus not clear if there is any dependence of the 𝑥 direction signal on 𝑧 forces.

4.3.2 Method by Wire

A second method of calibration was devised where a thin wire was used to impose a force on

the transducer in each direction separately up until breaking, assumed to occur at a consistent

load. With such a method, the load removal is effectively instantaneous, and since the forces

are low the effect of inertia of the force transducer on the measurements will be negligible.

Averages over 5 tests in each direction were computed, taking the step change in voltage from

the break event in order to find each row of the inverse of the calibration matrix of Equation 2.

Of course, the exact breaking force is not known, but setting the force vector to [1 1 1]T

allows calculation of the calibration matrix to some overall scale factor. The form of this scale

factor was then assumed by setting the 𝐶𝑧𝑧 element equal to the value found through the

aforementioned weight removal method. Thus, a calibration matrix was found and is shown in

Equation 3. The numeric values in the matrix have units mN pC⁄ , such that the charge amp

gain settings can be included. These are denoted 𝐴𝑖1 (units of V g⁄ ) and 𝐴𝑖2 (units of pC/g),

where 𝑖 = 𝑥, 𝑦, 𝑧.

Taking 𝐴𝑖 = 𝐴𝑖2 𝐴𝑖1⁄ , the form of Equation 3 is found (the dot-asterisk (.*) notation indicates

the multiplication is done by respective elements and not in a traditional matrix sense).

[

𝐶𝑥𝑥 𝐶𝑥𝑦 𝐶𝑥𝑧𝐶𝑦𝑥 𝐶𝑦𝑦 𝐶𝑦𝑧𝐶𝑧𝑥 𝐶𝑧𝑦 𝐶𝑧𝑧

] = [−180.26 2.86 2.11

1.97 −208.33 2.6631.39 −30.58 −405.23

] .∗ [

𝐴𝑥 𝐴𝑦 𝐴𝑧

𝐴𝑥 𝐴𝑦 𝐴𝑧

𝐴𝑥 𝐴𝑦 𝐴𝑧

] × 10−3 𝑁 𝑉⁄

Equation 3.

The calibration shows that the 𝑧 direction is twice as sensitive as the 𝑥 and 𝑦 directions, but

also that the 𝑧 direction is much more susceptible to cross-coupling.

22

4.4 MASS COMPENSATION

Mass compensation of the force transducer is required as it will pick up inertial forces due to

its own mass. To account for these forces and remove them so that contact forces may be

measured precisely, accelerations taken from each direction of the transducer faces (see Figure

3) are used in conjunction with effective masses of the device (decomposed into the 𝑥, 𝑦 and 𝑧

directions). It is assumed that the effects of rotational inertia on the measured forces is

negligible, and that the influence of off-axis acceleration on the measured forces can be

neglected. It is also assumed that the line of action of each accelerometer is coincident at the

centre of gravity of the force transducer.

4.4.1 First Estimate

A first estimate of the effective masses of the transducer was made by applying impulses to the

pin assembly to which it is attached, and respective forces and accelerations measured for each

direction. In this way, only inertial forces are measured by the transducer. Viewing the

calibrated responses in the frequency domain, and using data around a clear structural response,

the difference between force and acceleration magnitudes is a factor of the effective mass in

each direction (a similar averaging method was adopted as that used for accelerometer

calibration). Figure 14 shows calibrated force and acceleration signals, where an impulse was

applied to the side of the top hat dynamometer.

Figure 14. An example of the frequency domain signals used to find the mass compensation factors.

23

Finding the average scale factor between the 2 channels in the range 200 − 700 Hz gives an

effective 𝑥 direction mass of 17.9 g. The effective masses in each direction were taken as the

average over 3 separate impulse tests similar to that above. The calibration matrix was then

found to be:

[0.0184 0 0

0 0.0276 00 0 0.0125

] 𝑘𝑔

So that in order to find the true forces applied to the transducer (and noting that the force

measured due to �̈�, �̈� and �̈� is in the negative 𝑥, 𝑦 and 𝑧 directions respectively):

[

𝐹𝑋 𝑎𝑝𝑝𝑙𝑖𝑒𝑑

𝐹𝑌 𝑎𝑝𝑝𝑙𝑖𝑒𝑑

𝐹𝑍 𝑎𝑝𝑝𝑙𝑖𝑒𝑑

] = [𝐹𝑋 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

𝐹𝑌 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

𝐹𝑍 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑

] + [0.0184 0 0

0 0.0276 00 0 0.0125

] [�̈��̈��̈�

]

The fact that the effective mass is greater in the 𝑦 direction than the 𝑥 direction is surprising

given the symmetry of the transducer. This is likely to be due to details in the construction and

layup of the piezo crystals in the device.

4.4.2 Method of Least Squares

In order to check the reliability of this first mass calibration estimate, a second method was

tested. This involved taking the vectors of force and acceleration from each test (about a clear

peak again), concatenating them to create force and acceleration vectors of all tests, and then

using the method of least-squares to find an approximation to the effective mass matrix. The

least-squares fit of 𝐹 = 𝑀𝐴 has the form shown in Equation 4.

𝑀 = (𝐴𝐴𝑇)−1𝐹𝐴𝑇 Equation 4.

{ 𝐹 = [𝐹𝑥1

𝐹𝑦1

𝐹𝑧1

𝐹𝑥2

𝐹𝑦2

𝐹𝑧2

⋯……

𝐹𝑥𝑛

𝐹𝑦𝑛

𝐹𝑧𝑛

] ; 𝐴 = [�̈�1

�̈�1

�̈�1

�̈�2

�̈�2

�̈�2

⋯……

�̈�𝑛

�̈�𝑛

�̈�𝑛

] ; 𝑀 = [

𝑚𝑥𝑥 𝑚𝑥𝑦 𝑚𝑥𝑧

𝑚𝑦𝑥 𝑚𝑦𝑦 𝑚𝑦𝑧

𝑚𝑧𝑥 𝑚𝑧𝑦 𝑚𝑧𝑧

] }

This method gives the mass matrix shown in Equation 5 below:

𝑀 = [−16.8 6.8 0.0−1.3 −28.3 −3.4−2.9 −21.7 −18.1

] × 10−3 kg

Equation 5.

The diagonal entries show reasonable agreement with the first mass calibration method (noting

that the signs are the opposite due to the nature of the inertial forces), with the largest

discrepancy in the 3rd diagonal element. Off diagonal elements are mostly small, but there is a

24

strong dependency of the measured 𝑧 force on �̈�, stronger it appears than on �̈�. This is difficult

to interpret, and may be a consequence of the centripetal acceleration ≈ �̇�2 𝑟⁄ (and would

hence show frequency dependence), but is more likely due to the internal construction of the

device. From the system characterisation section, it was found that the out of plane (𝑥 direction)

transfer function showed a very different characteristic to the 𝑦 and 𝑧 direction transfer

functions, which indicates that the above method and in particular the off axis components may

be inaccurate. Some validation may still be gained through the diagonal components though,

and the process is illuminating to the degree of accuracy in the mass matrix elements used,

which will be valuable when performing future sensitivity studies.

25

5 BULK DATA ANALYSIS

When looking at individual cases of squeal and the limit cycles reached, it is important to put

the results in context with the bigger picture of squeal, which is what bulk data analysis enables.

By undergoing a series of repetitive tests under which several different regimes of squeal are

noted to occur, bulk data analysis helps to understand various general characteristics of squeal

events. These might be certain dominant frequencies inherent in the tests, experienced by either

the disc or the pin systems or common to both, or it could be a characteristic of the system to

switch squeal behaviour at a certain amplitude of vibration. Bulk post-processing also allows

short term dependencies of the test setup to be observed, for example a change in coefficient

of friction over a series of squeal tests, which might be interpreted as the consequence of a slow

temperature increase at the contact.

In performing large numbers of tests with different controllable parameters (preload, disc speed

and rotational direction here) as well as varying the initial conditions (for example “spinning-

up” the disc and bringing the pin into contact, or starting from still but connected), results of

the same type of test method may be compared to overall average behaviour, to show how these

state parameters influence the squeal behaviour.

Other observations were made during the tests, such as the inclination of a particular squeal

frequency originating from the disc to be fixed in space, and the notable reduction in the

propensity of the system to squeal at this frequency after the addition of the force transducer.

5.1 METHODS

5.1.1 Measurement

For the majority of tests performed for bulk data post-processing, measurement of squeal

parameters used 8 channels of data. These each had a sample rate of 50 kHz, giving a total of

400 kHz (the maximum allowed by the data acquisition card used). The channels measured

were:

1. Force transducer 𝑦 component

2. Force transducer 𝑧 component

3. Pin accelerometer 𝑦 component

4. Pin accelerometer 𝑧 component

5. Dynamometer 𝑦 component

6. Dynamometer 𝑧 component

7. Tachometer

8. Disc normal accelerometer

26

Low-pass filters were used as an anti-aliasing precaution on the force transducer and

accelerometer outputs (channels 1, 2, 3, 4 and 8), with a −3 dB point at 20 kHz.

The types of test that were performed included:

a) Continuous spinning (with continuous hand powered rotational input through the rotor)

with the pin initially in contact;

b) “Spinning up” the disc while the pin is not in contact then bringing it into contact.

Experimentation with forward and reverse rotation of the disc for the above two initial

conditions was also undertaken. For case a) the experimentalist is having to grasp the rotor, in

a fashion which is most likely unrepeatable, which will have an effect on the disc assembly

dynamics and hence a potential influence on the squeal frequencies.

Tests following the changing of a pin tip were noted, as were tests when the disc surface and

pin tip were cleaned. This was a necessity for cases of particularly long tests of sustained high

amplitude squeal, due to damage of the disc surface which may be caused. Such damage was

witnessed in the form of swarf carved from the disc as a consequence of particulates at the

contact patch causing particularly high friction forces.

The nature of the experimental setup – in particular of human control over rotation speed –

meant that tests performed at a controllable speed were not possible. However, measurement

of speed allows comparisons and dependence analyses on the parameter. Each test of the case

a) type was aimed to be undertaken at a reasonably constant speed, and large speed differences

reserved for separate tests.

All tests started with the disc accelerometer coincident with the tachometer, so that its position

is known at every point (given the test rotation direction).

27

5.1.2 Post-Processing

Data collected was post-processed using Matlab, in order to create a vector of desired

parameters for each period of test-time. This period was chosen to be 0.1 s for the results

presented in this report. The parameters generated were:

1. Mean velocity;

2. Mean friction and normal forces;

3. Mean coefficient of friction;

4. Force fluctuation ratios:

a. Friction direction;

b. Normal direction;

The mean velocity was calculated from the tachometer signal in the period of interest.

Calculation of the speed was performed at each rising/falling edge of the tachometer signal.

For slow tests involving squeal with significant tangential disc motion, rising and falling edges

are subject to electrical “bounce”. Hence precautions were taken in order to de-bounce the

tachometer signal, based on a feasible time over which the tachometer signal is likely to

fluctuate. This could be the same time over which a fast test measures genuine subsequent

rising and falling edges, so human judgement and knowledge of the test type was used to

effectively de-bounce the signal.

The mean friction and normal forces are the mean values of the calibrated dynamometer forces,

and the mean coefficient of friction = (mean friction force)/(mean normal force).

The force fluctuation ratios take the difference between the maximum and minimum values of

each calibrated force transducer signal and normalise it by the respective mean force. For the

friction force:

max(𝑦 𝑓𝑜𝑟𝑐𝑒𝑡𝑟𝑎𝑛𝑠𝑑𝑢𝑐𝑒𝑟) − min(𝑦 𝑓𝑜𝑟𝑐𝑒𝑡𝑟𝑎𝑛𝑠𝑑𝑢𝑐𝑒𝑟)

𝑦 𝑓𝑜𝑟𝑐𝑒𝑑𝑦𝑛𝑎𝑚𝑜𝑚𝑒𝑡𝑒𝑟̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅

The disc and pin dominant frequencies are the frequencies at which the highest, 2nd highest and

3rd highest peaks occur in the Discrete Fourier Transform (DFT) of the pin 𝑧 direction and the

disc transverse direction acceleration. The DFT finds the frequency content of the signal within

the window, but by doing so the sample is assumed to repeat in the time domain, and so the

last sample of the window is assumed to join with the first sample of the same window, which

is a repetition in the time domain. This can generate artificial frequency content, and so a

window function is required. A Blackman window function was chosen, and the product of the

5. Pin 𝑦 and transverse disc RMS accelerations;

6. Disc and pin dominant frequencies and peak

amplitudes;

7. Disc accelerometer angle (with respect to the

upright position);

8. Start and finish samples for each period.

28

function and the calibrated accelerations used as the input for the DFT. To find the peaks within

the DFT, smoothing was required such that a peak-search algorithm doesn’t pick up noise

variations on large peak edges and interpret them as strong peaks. This was implemented by

using a median filter, where the order of the median filter was taken as dependent on the

analysed window length and sample frequency. The peak-search algorithm was then used, with

a tuned minimum peak height and separation, in order to best represent the highest 3 individual

peaks in the frequency domain.

The disc accelerometer angle is simply calculated from the rising and falling edges of the

tachometer signal.

Although analyses presented here do not explore all interesting correlations between the

aforementioned parameters, they enabled a broader view of the parametric dependence for

refinement of presentation here, and the techniques used in their calculation will be a useful

tool for future departmental investigations into brake squeal.

5.1.3 Squeal Frequency Analysis

In order to gain a good picture of the dominant squeal frequencies, scatter plots of the dominant

frequencies of the disc and pin normal accelerations were created, and are shown in Figure 15,

16, 17 and 18. Figure 15 and 16 show results from 15 forward rotating tests, and Figure 17

and 18 from 9 backwards rotating tests. In order to understand how the dominant frequency

content might vary with amplitudes of vibration at the contact, the graphs are plotted against

the RMS amplitude of the pin 𝑦 direction acceleration. The centre of each plotted point gives

its true value, while the area of the plotted point is representative of the relative magnitude (in

points2 – where points here represent the smallest size in a Matlab scatter plot). The exact

relationship that gives the size of each point is:

𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 = √𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝐷𝐹𝑇 𝑝𝑒𝑎𝑘

𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝐷𝐹𝑇 𝑝𝑒𝑎𝑘 𝑖𝑛 𝑝𝑙𝑜𝑡 points

The experimentally measured resonances of the disc are shown by the black dot-dashed lines,

where each label gives: (𝑘, 𝑛) or (number of nodal diameters, number of nodal circles) of

the corresponding measured mode shape. The blue dot-dashed lines show the measured

resonances of the pin assembly.

All results with pin y RMS acceleration < 200 ms2 have been plotted faintly, as this data is

less likely to be representative of events exhibiting squeal.

29

Figure 15. Dominant frequencies measured by the disc mounted accelerometer in forwards rotating

tests. (largest peaks plotted in the background for clarity).

Figure 16. Dominant frequencies measured by the pin normal accelerometer in forwards rotating

tests. (largest peaks plotted in the background for clarity).

30

Figure 17. Dominant frequencies measured by the disc mounted accelerometer in backwards rotating

tests. (largest peaks plotted in the foreground for clarity).

Figure 18. Dominant frequencies measured by the pin normal accelerometer in backwards rotating

tests. (largest peaks plotted in the foreground for clarity).

31

Firstly, it is noted that dominant frequencies measured on the disc surface in the forward tests

mostly align well with disc resonances, or are harmonics of lower squeal frequencies. The pin

dominant frequencies correspond to the same strongest peaks in the disc, but fewer peaks align

with higher disc modes. The relative point areas in Figure 15 show that the amplitude of the

2.1 kHz component dominates the disc response at low pin 𝑦 RMS acceleration, where the

amplitude is so great that the peak finding algorithm – despite precautions on minimum peak

height and separation – has found peaks on the sidebands of this resonance. This dominance is

witnessed to a lesser extent in the corresponding pin plot. This regime of squeal is only

sustainable for lower pin accelerations, shown by the fact that this frequency component

becomes much less prevalent above a pin 𝑦 RMS acceleration of ≈ 900 ms−2 . This aligns

with the observation that the propensity of the system to squeal at this frequency was greatly

reduced by the force transducer addition. Greater forces are required to sustain this squeal

regime with the added mass, which could not be generated as easily with a new or clean pin

tip.

The second most prominent frequency is 1.25 kHz, and appears more stable across the whole

RMS acceleration range. Figure 16 shows a correlation between growth of DFT peak and RMS

acceleration, which suggests that the frequency is a pin system dependent one.

The disc frequencies in the backwards rotating direction are seen to fall again at natural

frequencies of the disc. However, the most prominent frequency now is 400 Hz, across the

entire pin RMS acceleration range. This regime of squeal behaviour is analysed in a case study

later. For backwards rotating squeal events, it appears that vibrational kinetic energy is spread

across frequencies more ubiquitously than for forwards rotating squeal, shown by the smaller

difference between plotted point areas. This was observed audibly during tests, where the

typical forwards rotating squeal had a clean but piercing tone, while typical reverse squeal was

noted to sound like many frequencies were present, and have a less piercing quality. The

maximum disc RMS acceleration measured during the forward tests was 4120 ms−2 compared

with 960 ms−2 from the backward tests, further substantiating these harshness observations.

It was also noted from the plots that the forward rotating tests showed more of the higher disc

modes than the reverse tests. The high amplitude disc vibration and corresponding kinetic

energy act to feed higher disc modes in a non-linear manner.

32

5.1.4 Fixed In Space Theory

The dominance of the 2.1 kHz frequency component in the forward rotating squeal

measurements was witnessed in the raw disc mounted accelerometer data as a “beating”

pattern. It was suggested that this beating was a consequence of the acclerometer passing

through consecutive nodes and anti-nodes. The disc mode closest to 2.1 kHz is the mode with

3 nodal diameters and 0 nodal circles (at 2076 Hz), and so the expected number of beating

envelopes per rotation is 6. Figure 19 shows plots of the raw accelerometer data, the same data

filtered using a bandpass filter centred at 2.1 kHz, and the speed of the disc as measured and

calculated from the tachometer signal shown in plots 𝑎, 𝑏 and 𝑐 respectively. The effect of

bandpass filtering the signal is to show clearly that the 2.1 kHz element dominates the signal,

and shows even clearer the progression through node and antinode in the period of sustained

squeal from 3.4 s to 7.2 s.

Figure 19. (a) Raw disc accelerometer signal during forwards rotating squeal (b) the same signal

bandpass filtered about the (k,n)=(3,0) disc resonance (c) the disc speed as measured by the

tachometer.

Figure 19(c) shows that the disc speed is very approximately constant over this range, and so

it is expected that the envelope width is consistent. The angle turned through in the 3 beats

between 4 s and 6.3 s is 180°, as expected. This finding shows that the conditions for vibration

at the contact point inducing this vibration must be steady. Analysis of this test and others

exhibiting forward rotating squeal show that the position of the contact point falls between a

33

node and an antinode of these steady oscillations. Again this is intuitive; it would not be

expected that the contact point falls on a node as this would correspond to the pin exhibiting

zero normal displacement, but the contact might also not be at an antinode as the pin may be

inhibiting to the natural motion of the disc – so an equilibrium point in between is reached.

This finding agrees with the conclusions of Gianini et al. [9].

34

6 CASE STUDIES

6.1 THE IMPORTANCE OF RADIAL FLUCTUATIONS

The pin structure is positioned in contact with the disc such that the steady friction force is as

close as possible to the 𝑦 direction of the pin. Due to the limits on height adjustment of the disc

and pin, the contact point is slightly elevated with respect to the centre of the disc (by ≈ 6 mm).

Radial force measurements have been made possible with the 3-axis transducer, and so the

importance of them on the contact force vector can be analysed.

Figure 20 shows a plot of tangential force against radial force of the pin given by the force

transducer during a 10 s test featuring typical steady reverse direction squeal, analysed in the

next section. It is clear that the radial force fluctuations are much smaller than the tangential

force fluctuations, also shown by the standard deviation of the radial force, at ≈ 11 times less

than the tangential force.

Figure 20. y (friction) force against x (radial) force measured by the force transducer

The radial pin displacements are commonly of the same order as the tangential displacements,

as shown by the graph of Figure 21 and supported by Table 4. The corresponding velocity and

acceleration data is also shown in Figure 22 and Figure 23, showing smaller radial fluctuations.

These data points are taken from the same test and use the same samples as the force fluctuation

analysis above. The comparable deflections in the 𝑥 direction despite relatively small forces

are due to the lower stiffness of the pin structure in this direction (the same low stiffness that

Table 3. Standard deviations of measured forces

FORCE COMPONENT

STANDARD DEVIATION (N)

X 1.88

Y 20.60

Z 48.65

35

is responsible for the lowest mode of the pin structure). The acceleration data shows a slight

correlation between 𝑥 and 𝑦, which is due to the small contact point offset mentioned above.

Figure 21. y vs x displacement fluctuations Figure 22. y vs x velocity fluctuations

Figure 23. y vs x acceleration fluctuations

STANDARD DEVIATIONS

DISPLACEMENT (𝝁𝒎)

VELOCITY

(𝒎𝒎𝒔−𝟏)

ACCELERATION

(𝒎𝒔−𝟐)

X 11.2 18.1 59.4 Y 28.8 86.7 415.4 Z 31.6 48.6 276.0

Table 4. Standard deviations of displacement, velocity and acceleration, measured over a 10s typical

backwards rotating squeal test

36

6.2 COMPARISONS WITH COULOMB FRICTION

The aim of re-instrumenting the force measurement method was to look in better detail at how

forces vary at the contact, and how motion is sustained by the contact forces in limit cycle

behaviour (or steady squealing), in order to compare friction models with experimental

measurements. Performing intuitive studies on the cyclic contact forces also helps inspire

confidence in the method of measurement and calibration, confidence which is essential for

more complex, less intuitive studies. Coulomb friction here refers to 𝐹 ≤ 𝜇𝑁 with a single

coefficient of friction.

In order to provide direction to the first analyses, the question was posed: “How close are the

measured friction forces to obeying Coulomb’s law?” If coulomb’s law is obeyed, then one

might assume that by plotting the measured friction force against the normal force in any test,

all data points should lie under a straight line which passes through the origin, where the

gradient of the line is the coefficient of friction. This is indeed a condition that needs to be

satisfied – the friction force cannot exceed a certain level at a certain normal force – but in

order to validate that it is the limit of friction that is expected, details of the motion at the

contact must be studied. Friction forces during sliding of the contact must satisfy 𝐹𝑟 = ±𝜇𝑁,

with respect to the sliding direction. The techniques used here may be used in the future to test

models which have a velocity dependent coefficient of friction.

Load cycles are mostly complex, typified by signals with a strong harmonic series, even more

so for squeal at higher preloads. This makes plots of friction and normal force difficult to

interpret. However, first intuitive analyses on 2 cases of low preload squeal are explored here,

which can form a starting point for future experimental contact force investigation.

6.2.1 Method

Channels of measurement included the 𝑦 and 𝑧 components of the force transducer, both

dynamometer outputs, 𝑦 and 𝑧 pin accelerometers, the tachometer and a tangentially mounted

disc accelerometer, shown Figure 24. By measuring the tangential acceleration of the disc,

relative displacements of the disc and pin may be calculated. However, the exact contact

accelerations will not be the same as those measured, since the pin 𝑦 accelerometer is mounted

on the side of the transducer, and the disc accelerometer on the reverse of the disc. The structure

of the pin assembly between the pin tip and the side of the transducer is not perfectly stiff, and

the effect of rotational acceleration of the pin assembly will be amplified between the

accelerometer position and the tip. Likewise with the disc accelerometer, transverse vibrations

37

will induce rotation at the disc surface and hence the disc accelerometer will not read the pure

tangential acceleration.

Figure 24. Accelerometer 4 mounted to measure tangential disc acceleration

Each test was performed over a rotation band of 22 tachometer disc holes (or 66°), with the

centre being the position at which the disc mounted accelerometer is exactly level with the pin,

between the 11th and 12th holes. A band of ±2.25° either side of the centre point was chosen

as an appropriate analysis band (the opening edge of the 11th hole and the closing edge of the

12th hole), over which integration of accelerations for velocities and displacements was

performed.

In order to find accurate forces, mixing of the force transducer and dynamometer signals was

required. This was done by designing a low pass 1st order Butterworth filter with −3 dB point

at 10 Hz for the dynamometer signals to extract the low frequency force components, and an

equivalent high pass filter with the same −3 dB point for the force transducer signals. Each

filter was used twice, in the forward and backwards directions, to ensure that the outputs have

zero phase shift. This does however mean that each designed −3 dB point is actually a −6 dB

point, causing a significant dip in overall output response at 10 Hz. This is acceptable for the

cases analysed, where the lowest (non-zero) frequency of interest is 140 Hz.

Cumulative trapezoidal numerical integration was used to find velocities and displacements

from the measured accelerations. This technique is susceptible to drift, so a similar method of

high pass filtering the signal was employed as for the force signals above, where the cut-off

frequency was chosen depending on the requirement of the particular signal not to drift;

typically 30 Hz. This method was verified with comparison to a first order Butterworth filter

integration approach, and deemed reliable upon comparison of the respective results; the filter

method showing similar outcomes but with greater dependence on the cut-off frequency

38

chosen. For the squeal regimes analysed, it was initially assumed that the effects of the angular

gradients in transverse disc displacement (𝜕𝑤

𝜕𝜃) – causing pivoting motion of the disc

accelerometer – were small, as the squeal frequencies measured were lower than any of the

transverse disc modes.

The steady state velocity of the disc was approximated as constant over the 4.5° analysis band,

and calculated as the arc length divided by the time window of the band. Finally, since only

the friction and normal components of force and acceleration were measured, the force

calibration matrix was reduced to just the 2nd and 3rd rows and columns, creating a 2x2 matrix

with no dependence on the 𝑥 direction forces. 𝑥 direction forces were found to be small in the

previous section, legitimising this simplification.

6.2.2 Backwards Rotating Squeal

The first example is for low preload squeal in the backwards rotation direction. Figure 25

shows how the contact forces vary over approximately one cycle, corresponding to 128

samples (a cycle frequency of 391 Hz). The arrows indicate the force path direction, where the

start of each arrow tail lies at sample 𝑁, and the tip of each arrow at sample (𝑁 + 1). Numbers

in the plot are sample numbers, which are shown as vertical lines in the plots of Figure 26. The

left plot shows the 𝑦 direction displacement of the disc and pin over the same cycle, and the

right plot shows the respective velocities. All three of these graphs have been plotted on top of

scattered data showing the bounds of error. This data was calculated by assuming that all of

the calibration constants found are subject to ±10% error. By applying a random error to each

calibration constant, and computing all accelerations and forces for an arbitrarily large number

of cases, the field of error shown in grey in each plot appears. 200 sets of randomly calibrated

data are plotted here.

The DFT of the 𝑦 direction pin and disc acceleration in the analysis band gives a fundamental

frequency component of 392 Hz. This was identified in the Bulk Data Analysis section as the

strongest and most common frequency measured during squeal events in the backwards

rotation direction, and is uncharacteristic of either of the separate systems. Figure 26 indicates

that tangential fluctuations of the pin displacement are much larger than corresponding disc

fluctuations, suggesting that the squeal frequency is strongly governed by the pin dynamics. A

similar conclusion was reached in the Bulk Data Analysis section since the peak values of the

392 Hz component in the DFT showed positive correlation with pin 𝑦 direction RMS

amplitude.

39

Figure 25. Friction vs Normal force for a low preload limit cycle during backwards rotating squeal.

Figure 26. Left (a): Pin and disc tangential displacements against sample number. Right (b): pin, disc

and relative velocities against sample number.

Figure 25 indicates that for ≈ 40% of the cycle, both forces are ~0. During this period (1st to

the 44th sample), the pin is seen to pass from its most negative to its most positive 𝑦

displacement. This is motion against the disc (illustrated in Figure 26(b) where the sliding

velocity – (𝑣𝑦,𝑑𝑖𝑠𝑐 − 𝑣𝑦,𝑝𝑖𝑛) – is negative), so a friction force would be expected if the bodies

40

were truly in contact and obeying Coulomb’s law. Since the friction force is negligibly small

during this window, the disc and pin are clearly not in contact.

The other 60% of the cycle features 3 sections which may be described as: normal force

loading, subsequent friction force loading, and simultaneous unloading. The region of samples

44 to 71 shows an increase in normal force, with a small change in the friction force, and the

velocity plot indicates a positive sliding velocity. If Coulomb’s law is satisfied under supposed

relative sliding, an extremely low friction coefficient ≈ −0.08 is found; this is clearly

unrealistically low, and “sticking” would be a more appropriate explanation here.

From sample 71 to 89, the pin velocity seemingly reverses, and the magnitude of the friction

force increases to a (directional) maximum ≈ −6.5 N. From sample 89 to 122 the forces drop

approximately linearly back to zero. In this region, the velocities of the pin and disc surface are

similar to the 44 to 71 sample band, and it might be assumed that Coulomb’s law is obeyed.

This time, a coefficient of ≈ −0.67 is observed. Clearly there are some disparities between

what one expects from measured forces, and what one expects from the measured velocities.

Accounting for the potential errors in measurement of velocities and displacements, it can be

seen that the measured pin acceleration could be an over estimation of the true value, and

sticking could feasibly be occurring. The sample range of 44 to 89 would be better matched to

the force plot under the assumption of sticking. Agreement with Coulomb would then be

satisfied, with a coefficient of friction of 𝜇 ≈ −0.67, and the overall cycle would be a case of

stick-slip.

Plotting friction against normal force over a 2 second squeal period yields the graph of Figure

27, which shows that this cycle is a repetitive, sustained one. Figure 28 shows a case for a

much higher preload and louder, more harsh squeal. The plot is very similar in shape to that of

Figure 25, and it can easily be seen why the harmonic content of the high preload case is greater

than that at low preload, as the cycle shows forces which vary in a much more erratic, less

rounded way.

41

Figure 27. Friction vs normal force for a period of 2 seconds during squeal, indicating the repetitive

nature of the limit cycle.

Figure 28. Friction vs normal force for high preload squeal in the backwards rotating; showing

likeness with the low preload case

42

6.2.3 Forwards Rotating Squeal

A second example illustrates a case where Coulomb friction may be a clearer conclusion.

Figure 29 shows one cycle of force for squeal resulting from forward motion of the disc. This

squeal could not be sustained for a long period, very quickly growing and dying away in a

period of ≈ 1 s. The plot shows a force cycle from just before the peak in friction force

magnitude. The arrows again make clear the direction of the force path, but for clarity, are

plotted over 3 samples (the tail at sample 𝑁, the tip at the sample (𝑁 + 3)). Friction follows a

linear relationship with normal force here, with only slight hysteresis in the cycle. A linear

regression has been plotted to show that the cycle shows a coefficient of friction of 𝜇 = 0.53,

with an intercept at 1.88 N. The grey background scatter was calculated as per the previous

section.

Figure 29. Friction vs normal force for low frequency squeal in the forwards rotation direction.

43

Figure 30. Left: Pin and disc tangential (y direction) displacements against sample number. Right:

pin, disc and relative velocities against sample number.

Figure 30 shows the tangential displacements of the disc and pin over one cycle on the left,

and tangential velocities on the right. Both figures highlight 2 samples ( 20 and 197 ),

corresponding approximately to the maximum and minimum friction (or normal) forces.

Performing a DFT on the disc and pin accelerometers shows that the fundamental squeal

frequency is 140 Hz. This is much lower than any pin or transverse disc mode, and was picked

up as a resonance in the transfer function of the coupled rotor and disc, in the system

characterisation section. Therefore tangential motion of the disc may be expected to be of more

influence to the results in this example, as is confirmed by the greater disc velocities shown in

the right of Figure 30. Pin and disc displacement and velocity fluctuations are approximately

in phase, and disc velocity fluctuations are measured as being much larger than the steady state

disc velocity. If this is the case, and also that the negative tangential velocity is larger than that

of the pin, the friction force would be expected to change sign, passing through zero. This is

not the case, and the minimum friction force measured is ≈ 11 N. By this token it appears again

that expectations from the force plot do not match expectations from the measured velocities,

and the reliability of the method of contact velocity measurement seems in jeopardy.

Since the 140 Hz disc assembly mode involved motion similar to the mode of the disc with 1

nodal diameter, it could well be the case that disc twisting or excessive off axis motion is

causing over-reading of the tangential disc accelerations and hence velocity, giving negative

sliding velocities when really this isn’t the case. It is clear from the force measurements that

oscillations do occur, but friction forces never reverse.

44

Both examples illustrate the scale of errors possible from the potential improper measurement

of contact accelerations (precautions of which were noted at the start of the measurement

section), and the effect of the respective assembly responses on these measurements. To be

able to measure the contact accelerations with enough precision to say that sticking or slipping

is occurring may require a greater degree of care and accuracy in calibration, and

experimentation with other – such as optical – methods could be beneficial. Each analysis

shows a different coefficient of friction under expected sliding conditions, of 𝜇 ≈ 0.67 and

𝜇 = 0.52 for backwards and forwards rotating incidences of squeal respectively.

6.3 LIMIT CYCLE BEHAVIOUR

Due to the 8 channel measurement limit of the data acquisition card used for experiments, it

was not possible to measure the transverse acceleration of the disc, as well as all of the channels

required for the force analyses above. Hence, a separate set of tests were performed with a

transverse disc accelerometer, attempting to measure similar regimes of squeal as the above

analyses, in order to find the dominant pin motion and how normal displacements correlate

with normal forces. Bearing in mind the caveat that was mentioned in the previous section

about the trustworthiness of the measured pin and disc accelerations, the results presented only

look to provide a qualitative view of the limit cycle motion.

The parameters measured were the pin and disc normal and tangential accelerations, and

tachometer signal. Similar methods described previously were used to find the velocities and

displacements.

The example analysed is that of higher preload, higher rotation speed, backwards rotating

squeal with a fundamental squeal frequency of 392 Hz (with force cycle shown in Figure 28).

The displacement of the pin in the 𝑦 − 𝑧 plane is shown in Figure 31, where arrows indicate

the direction of motion and their lengths give a gauge of the velocity (each arrow length

represents the distance travelled in 20 μs).

45

Figure 31. Limit cycle path in the y-z plane, constructed from pin accelerometer data.

Figure 32. Disc limit cycle path for one pin cycle, constructed from disc accelerometer data.

46

Figure 32 shows the displacements of the disc, where each arrow represents every 4 samples,

and so arrow lengths here are representative of the distance travelled in 80 μs . The

displacements of the pin are high pass filtered to prevent drift, and so the point of zero

displacement may not align with the origin of the resting position of the pin. The steady state

velocity of the disc has been included with the disc displacement plot, but again the normal

displacement is high pass filtered and so the origin may not align with the resting position.

The plots show that to a certain degree normal fluctuations are in phase. The extreme positive

normal displacements align at sample 37, and negative displacements align around samples

92 − 103. The disc appears to show a path reminiscent of a full wave rectified cosine, where

at sample 37 the disc “digs in” to the pin, while in the region 92 − 103 “free motion” might

be approximated. This aligns well with the knowledge gained from the first case study. The

sample range from 131 − 97 (via 37) aligns approximately with the sample range of 44 −

122 from the aforementioned analysis, upon comparison with the displacements plotted in

Figure 26. This is the period of the cycle where significant forces are generated, and it was

concluded before that the measured pin friction acceleration – and hence displacement – may

be greater than its true value. The loop which sample 37 forms a part of is then not

representative, but this comparison does substantiate the postulate of free motion at the extreme

of negative disc normal displacement.

The intuitive nature of the above analyses has highlighted the requirement for more accurate

methods of acceleration, velocity or displacement measurement, in order to investigate friction

models resulting from different types of squeal regime. The force transducer has enabled

intricate details of the contact forces to be measured, and coupled with the DC components

from the dynamometer a good basis for future contact force investigations has been set.

47

7 CONCLUSIONS

An experimental investigation of disc brake squeal was undertaken using a reduced pin-on-

disc test rig, inherited from previous departmental investigations into the phenomenon. With

respect to the aims outlined at the start of the project:

Instrumentation was added to the test rig, consisting of a 3-axis force transducer,

mounted on the top hat dynamometer used previously for force measurement. A

calibration matrix was found, and mass compensation factors used with calibrated

accelerometers in order to find accurate contact forces between the pin and disc;

The pin and disc assemblies were thoroughly characterised, using analytical

calculations and experimental measurements, and compared with FEM calculations

performed by Edmondson-Jones [15]. All experimental disc modes with zero nodal

circles were found to be in approximate agreement with the FEA.

Bulk data analysis was undertaken, and post-processing software developed in order to

analyse certain characteristics of squeal events. The dominant frequencies of squeal

were found to be 2.1 kHz in the forward rotating direction, and 400 Hz in the

backwards rotating direction. The dominant 2.1 kHz squeal was found to coincide with

the disc mode of 3 nodal diameters and 0 nodal circles, and analysis of tests exhibiting

this squeal in the time domain showed that the disc was indeed exhibiting this mode

and that the mode shape was fixed in space (in agreement with Giannini et al. [9]).

Relative magnitudes of dominant frequency components were compared, and revealed

– alongside observation – that harsh squeal results from energy being concentrated at a

particular frequency, showed in the differences between forward and backward rotating

tests.

Off-axis (radial) fluctuations were measured and shown to have a standard deviation of

force ≈ 10 times lower than that in the frictional direction, while radial acceleration

fluctuations showed a standard deviation ≈ 7 times lower than the frictional direction,

validating the assumption that radial fluctuations are small.

2 Case studies were explored with the aim of analysing the limit cycle behaviour of the

system during squeal, in the forwards and backwards rotating directions. The

backwards rotating case study showed that loss of contact, sticking and slipping may

all be occurring within the limit cycle. This was corroborated by a period of friction

and normal force ≈ 0 𝑁 for 40% of the cycle, a subsequent normal force loading,

48

followed by a friction force loading. Simultaneous unloading then gives a linear slope

comparable with Coulomb’s law. The second example showed strongly linear

behaviour between friction and normal forces for a case of low frequency forward

rotating squeal, where a constant sliding velocity is predicted. Attempted measurement

of contact velocities and displacements showed poor agreement with the forces

measured, and updated methods for acceleration, velocity and displacement

measurement suggested.

8 APPENDIX

8.1 RISK ASSESSMENT

The risk assessment made at the start of the project indicated the main hazard as the squeal

volume causing discomfort. This was true, but squeal was so loud at times that others in the

lab required ear defenders as well. A risk assessment with more attention paid to others

would have been beneficial.

49

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